# Three Puzzles from Searchers

Some people reach this blog with search engines. But what do they search for?

Some are still looking for the answers to that 24 puzzle that I got wrong. Deal or No Deal still generates hits. Lots want UFT information. (More about this later this week). Some are really looking for other bloggers I have mentioned or I link to.

But some are searching for puzzles that I haven't written about.

Here are, as best as I can construct, 3 searcher-driven puzzles:

**1, 3, 4, 6 make 23**

I get it. Combine these 4 numbers using +, -, *, and / to make 23. I don't see a solution without making up new rules. Readers?

**Largest 9 digit perfect square no 9s.**

Seems to have clear ground rules. I know that 20,000^{2} = 400,000,000, but I assume we can do much better. But I can't imagine checking the next 9,999 perfect squares.

As an aside, the number of 9 digit numbers is 900,000,000. The number without the digit '9' is less than 350,000,000. Even stripping away the last hundred million, a random choice has a greater than 50% probability of including a '9'.

**Largest 3 digit number 10 factors**

OK, this seems clear, too. I can step in on this one, but let's see if a reader wants to attack it first.

Keep solving and keep searching!

The second one is suceptible to attack by computer, but I can’t see any other reasonable way of doing it. Since 30,000^2 = 900,000,000 is the first which we know certainly fails, it might be more fruitful to start at the top and work down, since then at least once you get one you know you’re done.

Mathematica tells me that 7 of the first 9 squares, 58 of the next 90, 474 of the next 900, and 3919 of the next 9000 have no nines in them, as do 4161 of the squares between 20,000^2 and 30,000^2. The largest of these is 29812^2 = 888755344.

Ah, the advantage of Mathematica over my primitive spreadsheets….

I was wondering how far you could reasonably travel narrowing down candidates. Nothing that ends in 3 or 7 (brings us down to 8000 possibilities). But the rest of my tricks take out less than 20% of the remains. Even 3 very good such tricks would leave us over 3000… need the computer for this as far as I can tell.

The third didn’t interest you?

Well, once we know what the answer to the second one is, there were really only 200 things to check (starting from the top) and cutting out those ending in 3 and 7 brings it down to 160. So, it would have been doable in “not too long,” although there wasn’t any way you could have known that in advance.

Actually! Here’s a clever thought that would have gotten it in 3 calculations: I said we should start from the top, since 900,000,000 =s 30,000^2 is the first thing we know fails. In fact, though, anything larger than 888,888,888 certainly fails (it’ll either have at least one 9 or be 10 digits). So we can just take the square root there — that gives us 29814.23…. So our possibilities must be 29814 or less. 29814^2 = 888874596 has a 9. 29813^2 ends in 9. And 29812 works. So 3 calculations total to get the answer. Of course, we couldn’t guarantee in advance that this process would work so quickly, but since it does, I think I can still give myself a pat on the back. :-)

The third is nice — we need to find the largest prime less than 1000/16 = 62.5 and 1000/81 = 12.3… (1000/625

Hmm, do you have a character limit? It seems the end of my last comment was cut off.

Continuing:

(1000/625

ack! It must not like my use of the “less than” sign.

1000/625 is less than 2, so that’s of no use to us. Then we need to compare 16*61, 81*11 and 2^9. Of course, I haven’t said *why* we need to do any of this.

That was great. I thought you stopped at 1000/625 since it was so clearly useless.

Yes, my comments will interpret those less thans as a left container and assume you are writing (bad) html.

For the first one, how about 6*3 + 4 + 1? (Assuming the rules are that you must use each number precisely once and each operation zero or more times.)

Looks good Nick.

All I could think of was 46/(3 – 1) , which seems like cheating.